1 Introduction

This experiment investigates whether different foodstuffs (A, B, C, and D) produce different average weights.

The experiment follows a Completely Randomized Design (CRD) where foodstuff type is the treatment factor and weight is the response variable.

2 Objectives

Summarize the data descriptively.
Visualize the weight distributions.
Test whether mean weights differ among foodstuffs.
Conduct multiple comparisons.
Check ANOVA assumptions.
Draw conclusions.

3 Data Entry

Weight <- c(
55,49,42,21,52,
61,112,30,89,63,
42,97,81,95,92,
169,137,169,85,154
)

Foodstuff <- factor(c(
rep("A",5),
rep("B",5),
rep("C",5),
rep("D",5)
))

food <- data.frame(Foodstuff, Weight)

food

##    Foodstuff Weight
## 1          A     55
## 2          A     49
## 3          A     42
## 4          A     21
## 5          A     52
## 6          B     61
## 7          B    112
## 8          B     30
## 9          B     89
## 10         B     63
## 11         C     42
## 12         C     97
## 13         C     81
## 14         C     95
## 15         C     92
## 16         D    169
## 17         D    137
## 18         D    169
## 19         D     85
## 20         D    154

4 Structure of Data

str(food)

## 'data.frame':    20 obs. of  2 variables:
##  $ Foodstuff: Factor w/ 4 levels "A","B","C","D": 1 1 1 1 1 2 2 2 2 2 ...
##  $ Weight   : num  55 49 42 21 52 61 112 30 89 63 ...

5 Descriptive Statistics

5.1 Sample Sizes

table(food$Foodstuff)

## 
## A B C D 
## 5 5 5 5

5.2 Means

aggregate(Weight ~ Foodstuff,
          data = food,
          mean)

##   Foodstuff Weight
## 1         A   43.8
## 2         B   71.0
## 3         C   81.4
## 4         D  142.8

5.3 Standard Deviations

aggregate(Weight ~ Foodstuff,
          data = food,
          sd)

##   Foodstuff   Weight
## 1         A 13.62718
## 2         B 31.02418
## 3         C 22.87575
## 4         D 34.90272

5.4 Summary Statistics

aggregate(Weight ~ Foodstuff,
          data = food,
          summary)

##   Foodstuff Weight.Min. Weight.1st Qu. Weight.Median Weight.Mean Weight.3rd Qu.
## 1         A        21.0           42.0          49.0        43.8           52.0
## 2         B        30.0           61.0          63.0        71.0           89.0
## 3         C        42.0           81.0          92.0        81.4           95.0
## 4         D        85.0          137.0         154.0       142.8          169.0
##   Weight.Max.
## 1        55.0
## 2       112.0
## 3        97.0
## 4       169.0

6 Exploratory Data Analysis

6.1 Boxplot

boxplot(Weight ~ Foodstuff,
        data = food,
        col = c("lightblue","lightgreen",
                "lightyellow","pink"),
        main = "Weight by Foodstuff",
        xlab = "Foodstuff",
        ylab = "Weight")

6.2 Dot Plot

stripchart(Weight ~ Foodstuff,
           data = food,
           vertical = TRUE,
           method = "jitter",
           pch = 19,
           col = "blue")

6.3 Means Plot

means <- aggregate(Weight ~ Foodstuff,
                   data = food,
                   mean)

barplot(means$Weight,
        names.arg = means$Foodstuff,
        ylab = "Mean Weight",
        main = "Mean Weight by Foodstuff")

7 Completely Randomized Design Model

The CRD model is

\[ Y_{ij} = \mu + \tau_i + \epsilon_{ij} \]

where

\(Y_{ij}\) = observed weight
\(\mu\) = overall mean
\(\tau_i\) = effect of foodstuff
\(\epsilon_{ij}\) = random error

8 Hypotheses

\[ H_0:\mu_A=\mu_B=\mu_C=\mu_D \]

\[ H_a:\text{At least one mean differs} \]

9 One-Way ANOVA

model <- aov(Weight ~ Foodstuff,
             data = food)

summary(model)

##             Df Sum Sq Mean Sq F value   Pr(>F)    
## Foodstuff    3  26235    8745   12.11 0.000218 ***
## Residuals   16  11559     722                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

10 ANOVA Table

anova(model)

## Analysis of Variance Table
## 
## Response: Weight
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## Foodstuff  3  26235  8745.0  12.105 0.000218 ***
## Residuals 16  11559   722.4                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

11 Treatment Means

model.tables(model,
             type = "means")

## Tables of means
## Grand mean
##       
## 84.75 
## 
##  Foodstuff 
## Foodstuff
##     A     B     C     D 
##  43.8  71.0  81.4 142.8

12 Multiple Comparisons

TukeyHSD(model)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = Weight ~ Foodstuff, data = food)
## 
## $Foodstuff
##     diff       lwr       upr     p adj
## B-A 27.2 -21.43481  75.83481 0.4061728
## C-A 37.6 -11.03481  86.23481 0.1621713
## D-A 99.0  50.36519 147.63481 0.0001380
## C-B 10.4 -38.23481  59.03481 0.9268359
## D-B 71.8  23.16519 120.43481 0.0032490
## D-C 61.4  12.76519 110.03481 0.0112771

13 Confidence Intervals for Treatment Means

tapply(food$Weight,
       food$Foodstuff,
       mean)

##     A     B     C     D 
##  43.8  71.0  81.4 142.8

14 Assumption Checking

14.1 Residual Plots

par(mfrow=c(2,2))
plot(model)

15 Normality of Residuals

15.1 Histogram

hist(residuals(model),
     main="Histogram of Residuals",
     xlab="Residuals")

15.2 Q-Q Plot

qqnorm(residuals(model))
qqline(residuals(model))

15.3 Shapiro-Wilk Test

shapiro.test(residuals(model))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(model)
## W = 0.93614, p-value = 0.2025

Interpretation:

p-value > 0.05 → residuals approximately normal.
p-value < 0.05 → residuals not normally distributed.

16 Homogeneity of Variance

bartlett.test(Weight ~ Foodstuff,
              data = food)

## 
##  Bartlett test of homogeneity of variances
## 
## data:  Weight by Foodstuff
## Bartlett's K-squared = 3.1571, df = 3, p-value = 0.368

Interpretation:

p-value > 0.05 → equal variances.
p-value < 0.05 → unequal variances.

17 Manual ANOVA Calculations

17.1 Grand Mean

grand.mean <- mean(food$Weight)
grand.mean

## [1] 84.75

17.2 Total Sum of Squares

SST <- sum((food$Weight - grand.mean)^2)
SST

## [1] 37793.75

17.3 Treatment Sum of Squares

means <- tapply(food$Weight,
                food$Foodstuff,
                mean)

r <- 5

SSTR <- sum(r*(means-grand.mean)^2)

SSTR

## [1] 26234.95

17.4 Error Sum of Squares

SSE <- SST - SSTR

SSE

## [1] 11558.8

17.5 Degrees of Freedom

df.treatment <- 4 - 1
df.error <- 20 - 4

c(df.treatment, df.error)

## [1]  3 16

17.6 Mean Squares

MST <- SSTR/df.treatment
MSE <- SSE/df.error

c(MST,MSE)

## [1] 8744.983  722.425

17.7 F Statistic

F.value <- MST/MSE
F.value

## [1] 12.10504

17.8 P-value

pf(F.value,
   df.treatment,
   df.error,
   lower.tail = FALSE)

## [1] 0.0002180248

18 Treatment Means Table

aggregate(Weight ~ Foodstuff,
          data=food,
          function(x)
            c(
              Mean=mean(x),
              SD=sd(x),
              Min=min(x),
              Max=max(x)
            ))

##   Foodstuff Weight.Mean Weight.SD Weight.Min Weight.Max
## 1         A    43.80000  13.62718   21.00000   55.00000
## 2         B    71.00000  31.02418   30.00000  112.00000
## 3         C    81.40000  22.87575   42.00000   97.00000
## 4         D   142.80000  34.90272   85.00000  169.00000

19 Interpretation

Interpret the ANOVA results:

If p-value < 0.05, conclude that foodstuff significantly affects weight.
If p-value > 0.05, conclude that there is no significant evidence of foodstuff differences.

Interpret Tukey’s test to identify which foodstuffs differ significantly.

20 Conclusions

Based on the analyses:

Compare mean weights among foodstuffs.
State whether significant differences exist.
Identify the best-performing foodstuff.
Comment on the ANOVA assumptions.

21 Session Information

sessionInfo()

## R version 4.3.3 (2024-02-29 ucrt)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19045)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=English_United States.utf8 
## [2] LC_CTYPE=English_United States.utf8   
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.utf8    
## 
## time zone: Africa/Nairobi
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## loaded via a namespace (and not attached):
##  [1] digest_0.6.37     R6_2.6.1          fastmap_1.2.0     xfun_0.52        
##  [5] cachem_1.1.0      knitr_1.51        htmltools_0.5.8.1 rmarkdown_2.31   
##  [9] lifecycle_1.0.4   cli_3.6.2         sass_0.4.9        jquerylib_0.1.4  
## [13] compiler_4.3.3    rstudioapi_0.17.1 tools_4.3.3       evaluate_1.0.4   
## [17] bslib_0.9.0       yaml_2.3.10       rlang_1.1.5       jsonlite_2.0.0

Completely Randomized Design (CRD): Effect of Foodstuff on Weight

Dr. Tum Isaac

2026-06-11